Finding a variogram model is one of most important and often difficult tasks in geostatistics/property modeling as it identifies the maximum and minimum directions of continuity of a given geologic or petrophysical property or any spatially correlated property. The “maximum direction of continuity” is the azimuth along which the variance of a given property changes the least. The “minimum direction of continuity” is a direction perpendicular to the maximum direction of continuity, which is the azimuth along which the variance of a given property changes the most.
Conventional methods for the computation and fitting of a traditional semi-variogram often require domain expertise on the part of the user and considerable trial and error. Conventional methods for automated semi-variogram fitting also focus on least squares methods of fitting a curve to a set of points representing an experimental semi-variogram.
Many commercial software packages offer traditional trial and error fitting. In FIG. 1, for example, traditional trial and error semi-variogram modeling is illustrated using ten (10) experimental semi-variograms in a graphical user interface 100. Each experimental semi-variogram is computed along a different azimuth. The number of experimental semi-variograms is dependent on the number of input data points and the number of data pairs in the computation. Ten were chosen for this example and produced satisfactory results based on 261 input data points. The user must experiment with the number of direction, with a minimum of 2 and a maximum of 36; the latter of which is computed every 5 degrees.
In each semi-variogram illustrated in FIG. 1, the user drags a vertical line 102 (left or right) using a pointing device until a line 104 is a “best fit” between the points in each semi-variogram. The user also has a choice of model types such as, for example, spherical, exponential, and Gaussian, when fitting the experimental semi-variogram points. This type of non-linear fitting is available in commercial software packages, such as a public domain product known as “Uncert,” which is a freeware product developed by Bill Wingle, Dr. Eileen Poeter, and Dr. Sean McKenna.
In automated fitting, the concept would also be to fit a curve to the semi-variogram points, but the software would use some approximation of the function to produce the best fit. As illustrated in FIG. 2, for example, traditional automated-linear semi-variogram fittings are compared to each experimental semi-variogram for FIG. 1 in the display 200. The linear best-fit shown in FIG. 2, however, is not very good for most rigorous cases. In most automated cases, the approach requires some form of curve (non-linear) fitting method that is “blind” to the user. An approach is blind to the user when the user cannot give any input to the fit achieved by the automated function.
There is, therefore, a need for a variogram model that enables non-linear semi-variogram fitting, is not blind to the user and can be automated.